Group #31807254 — subgroups

Subgroups — #31807254

Order 17792.

312 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 97 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22580977	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#17984761	C₂	1	2	yes	maximal, complemented
3	C₂×#17984759	C₂	1	2	yes	maximal
4	C₄×#14555769	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#124×C₁₃₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#14555769	C₂²	1	2	yes	—
7	C₂×#14555769	C₂²	1	2	yes	—
8	C₄⋊(C₁₃₉⋊C₈)	—	2	2	no	—
9	#17984761	—	2	4	no	—
10	#17984759	—	2	4	no	—
11	#17984753	—	2	1	no	—
12	C₄×C₁₃₉⋊C₈	—	2	1	no	—
13	C₂²×Q₈×C₁₃₉	C₂²	1	1	yes	characteristic
14	C₂²×D₄×C₁₃₉	C₂²	1	1	yes	characteristic
15	#124×C₁₃₉	—	2	2	no	—
16	#124×C₁₃₉	—	2	1	no	—
17	#124×C₁₃₉	—	2	1	no	—
18	C₂×C₂²⋊C₄×C₁₃₉	C₄	1	4	yes	—
19	C₂×C₄²×C₁₃₉	C₂²	1	1	yes	characteristic
20	#14555769	—	2	2	no	—
21	#14555769	—	2	2	no	—
22	#14555769	—	2	2	no	—
23	#14555769	D₄	1	4	yes	—
24	C₂×C₁₃₉⋊C₈	—	2	2	no	—
25	C₂×C₁₃₉⋊C₈	—	2	2	no	—
26	C₂⁴×C₁₃₉	C₂×C₄	1	2	yes	—
27	C₂×Q₈×C₁₃₉	—	2	1	no	—
28	C₂×Q₈×C₁₃₉	—	2	1	no	—
29	C₂×Q₈×C₁₃₉	—	2	2	no	—
30	C₂×Q₈×C₁₃₉	—	2	2	no	—
31	C₂×D₄×C₁₃₉	—	2	1	no	—
32	C₂×D₄×C₁₃₉	—	2	1	no	—
33	C₂×D₄×C₁₃₉	—	2	2	no	—
34	C₂×D₄×C₁₃₉	—	2	2	no	—
35	C₂²×C₄×C₁₃₉	C₂×C₄	1	2	yes	—
36	C₂²×C₄×C₁₃₉	C₂×C₄	1	2	yes	—
37	C₂²×C₄×C₁₃₉	C₂³	1	1	yes	characteristic
38	C₂²⋊C₄×C₁₃₉	—	2	4	no	—
39	C₂²⋊C₄×C₁₃₉	—	2	4	no	—
40	C₄²×C₁₃₉	D₄	1	2	yes	—
41	C₄²×C₁₃₉	D₄	1	1	yes	characteristic
42	C₄²×C₁₃₉	D₄	1	1	yes	characteristic
43	C₁₃₉⋊C₈	—	2	4	no	—
44	C₁₃₉⋊C₈	—	4	2	no	—
45	C₂³×C₁₃₉	—	2	2	no	—
46	C₂³×C₁₃₉	—	2	4	no	—
47	C₂³×C₁₃₉	—	4	4	no	—
48	C₂³×C₁₃₉	C₂²×C₄	1	1	yes	characteristic
49	Q₈×C₁₃₉	—	4	2	no	—
50	Q₈×C₁₃₉	—	4	1	no	—
51	Q₈×C₁₃₉	—	4	1	no	—
52	D₄×C₁₃₉	—	4	2	no	—
53	D₄×C₁₃₉	—	4	1	no	—
54	D₄×C₁₃₉	—	4	1	no	—
55	C₂×C₄×C₁₃₉	—	2	2	no	—
56	C₂×C₄×C₁₃₉	—	2	4	no	—
57	C₂×C₄×C₁₃₉	—	4	2	no	—
58	C₂×C₄×C₁₃₉	C₂²⋊C₄	1	2	yes	—
59	C₂×C₄×C₁₃₉	C₂²⋊C₄	1	2	yes	—
60	C₂×C₄×C₁₃₉	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄×C₁₃₉	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄×C₁₃₉	#41	1	2	yes	—
64	C₂²×C₁₃₉	—	2	2	no	—
65	C₂²×C₁₃₉	—	4	2	no	—
66	C₂²×C₁₃₉	—	4	4	no	—
67	C₂²×C₁₃₉	—	8	4	no	—
68	C₂²×C₁₃₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₁₃₉	C₄×D₄	1	2	yes	—
70	C₄×C₁₃₉	—	2	2	no	—
71	C₄×C₁₃₉	—	2	2	no	—
72	C₄×C₁₃₉	—	4	2	no	—
73	C₄×C₁₃₉	#121	1	2	yes	—
74	C₄×C₁₃₉	#124	1	1	yes	characteristic
75	C₄×C₁₃₉	#127	1	1	yes	characteristic
76	C₂×C₁₃₉	—	2	2	no	—
77	C₂×C₁₃₉	—	8	2	no	—
78	C₂×C₁₃₉	#390	1	1	yes	characteristic
79	C₂×C₁₃₉	#413	1	2	yes	—
80	C₁₃₉	#1967	1	1	yes	characteristic, complemented, Sylow
81	#1967	—	139	1	no	maximal, Sylow
82	C₂×#124	D₁₃₉	1	1	yes	characteristic
83	#423	—	139	1	no	—
84	C₂×#101	—	139	2	no	—
85	C₂×#100	—	139	2	no	—
86	C₄×M₁₆	—	139	1	no	—
87	C₂²×Q₈	C₂×D₁₃₉	1	1	yes	characteristic
88	C₂²×D₄	C₂×D₁₃₉	1	1	yes	characteristic
89	C₂×M₁₆	—	139	2	no	—
90	C₂×M₁₆	—	139	2	no	—
91	#124	—	2	1	no	—
92	#124	—	2	1	no	—
93	#124	—	2	2	no	—
94	C₂×C₂²⋊C₄	Dic₁₃₉	1	4	yes	—
95	C₂×C₄²	C₂×D₁₃₉	1	1	yes	characteristic
96	C₄⋊C₈	—	278	2	no	—
97	#101	—	278	4	no	—
98	#100	—	278	4	no	—
99	#97	—	278	1	no	—
100	C₄×C₈	—	278	1	no	—
101	C₂⁴	C₂×Dic₁₃₉	1	2	yes	—
102	C₂×Q₈	—	2	2	no	—
103	C₂×Q₈	—	2	1	no	—
104	C₂×Q₈	—	2	1	no	—
105	C₂×Q₈	—	2	2	no	—
106	C₂×D₄	—	2	2	no	—
107	C₂×D₄	—	2	1	no	—
108	C₂×D₄	—	2	1	no	—
109	C₂×D₄	—	2	2	no	—
110	C₂²×C₄	C₂×Dic₁₃₉	1	2	yes	—
111	C₂²×C₄	C₂×Dic₁₃₉	1	2	yes	—
112	C₂²×C₄	C₂²×D₁₃₉	1	1	yes	characteristic
113	M₁₆	—	278	2	no	—
114	M₁₆	—	278	2	no	—
115	M₁₆	—	278	2	no	—
116	M₁₆	—	139	4	no	—
117	C₂×C₈	—	278	2	no	—
118	C₂×C₈	—	278	2	no	—
119	C₂²⋊C₄	—	2	4	no	—
120	C₂²⋊C₄	—	2	4	no	—
121	C₄²	C₂²⋊D₁₃₉	1	1	yes	characteristic
122	C₄²	C₂²⋊D₁₃₉	1	1	yes	characteristic
123	C₄²	C₂²⋊D₁₃₉	1	2	yes	—
124	C₂³	—	2	2	no	—
125	C₂³	—	2	4	no	—
126	C₂³	—	4	4	no	—
127	C₂³	C₂²×Dic₁₃₉	1	1	yes	characteristic
128	Q₈	—	4	2	no	—
129	Q₈	—	4	1	no	—
130	Q₈	—	4	1	no	—
131	D₄	—	4	2	no	—
132	D₄	—	4	1	no	—
133	D₄	—	4	1	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	4	no	—
136	C₂×C₄	—	4	2	no	—
137	C₂×C₄	C₂²⋊Dic₁₃₉	1	2	yes	—
138	C₂×C₄	C₂²⋊Dic₁₃₉	1	2	yes	—
139	C₂×C₄	D₄×D₁₃₉	1	2	yes	—
140	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
141	C₂×C₄	C₂×C₂²⋊D₁₃₉	1	1	yes	characteristic
142	C₂×C₄	#14555784	1	2	yes	—
143	C₈	—	278	4	no	—
144	C₈	—	556	2	no	—
145	C₂²	—	2	2	no	—
146	C₂²	—	4	2	no	—
147	C₂²	—	4	4	no	—
148	C₂²	—	8	4	no	—
149	C₂²	C₂×C₂²⋊Dic₁₃₉	1	1	yes	characteristic
150	C₂²	D₄×Dic₁₃₉	1	2	yes	—
151	C₄	—	2	2	no	—
152	C₄	—	2	2	no	—
153	C₄	—	4	2	no	—
154	C₄	#17984813	1	2	yes	—
155	C₄	#17984829	1	1	yes	characteristic
156	C₄	#17984838	1	1	yes	characteristic
157	C₂	—	2	2	no	—
158	C₂	—	8	2	no	—
159	C₂	#22580856	1	1	yes	characteristic
160	C₂	#22580937	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22580977

C₂

yes

maximal, characteristic, complemented

C₂×#17984761

C₂

yes

maximal, complemented

C₂×#17984759

C₂

yes

maximal

C₄×#14555769

C₂

yes

maximal, characteristic, complemented

C₂×#124×C₁₃₉

C₂

yes

maximal, characteristic

C₂×#14555769

C₂²

yes

—

C₂×#14555769

C₂²

yes

—

C₄⋊(C₁₃₉⋊C₈)

—

#17984761

—

#17984759

—

#17984753

—

C₄×C₁₃₉⋊C₈

—

C₂²×Q₈×C₁₃₉

C₂²

yes

characteristic

C₂²×D₄×C₁₃₉

C₂²

yes

characteristic

#124×C₁₃₉

—

#124×C₁₃₉

—

#124×C₁₃₉

—

C₂×C₂²⋊C₄×C₁₃₉

C₄

yes

—

C₂×C₄²×C₁₃₉

C₂²

yes

—

—

—

D₄

yes

—

C₂×C₁₃₉⋊C₈

—

C₂×C₁₃₉⋊C₈

—

C₂⁴×C₁₃₉

C₂×C₄

yes

—

C₂×Q₈×C₁₃₉

—

C₂×Q₈×C₁₃₉

—

C₂×Q₈×C₁₃₉

—

C₂×Q₈×C₁₃₉

—

C₂×D₄×C₁₃₉

—

C₂×D₄×C₁₃₉

—

C₂×D₄×C₁₃₉

—

C₂×D₄×C₁₃₉

—

C₂²×C₄×C₁₃₉

C₂×C₄

yes

—

C₂²×C₄×C₁₃₉

C₂×C₄

yes

—

C₂²×C₄×C₁₃₉

C₂³

yes

characteristic

C₂²⋊C₄×C₁₃₉

—

C₂²⋊C₄×C₁₃₉

—

C₄²×C₁₃₉

D₄

yes

—

C₄²×C₁₃₉

D₄

yes

characteristic

C₄²×C₁₃₉

D₄

yes

characteristic

C₁₃₉⋊C₈

—

C₁₃₉⋊C₈

—

C₂³×C₁₃₉

—

C₂³×C₁₃₉

—

C₂³×C₁₃₉

—

C₂³×C₁₃₉

C₂²×C₄

yes

characteristic

Q₈×C₁₃₉

—

Q₈×C₁₃₉

—

Q₈×C₁₃₉

—

D₄×C₁₃₉

—

D₄×C₁₃₉

—

D₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

—

C₂×C₄×C₁₃₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

—

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₉

#41

yes

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

—

C₂²×C₁₃₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₃₉

C₄×D₄

yes

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

—

C₄×C₁₃₉

#121

yes

—

C₄×C₁₃₉

yes

C₄×C₁₃₉

yes

C₂×C₁₃₉

—

C₂×C₁₃₉

—

C₂×C₁₃₉

#390

yes

characteristic

C₂×C₁₃₉

#413

yes

—

C₁₃₉

#1967

yes

characteristic, complemented, Sylow

#1967

—

139

maximal, Sylow

C₂×#124

D₁₃₉

yes

characteristic

#423

—

139

—

C₂×#101

—

139

—

C₂×#100

—

139

—

C₄×M₁₆

—

139

—

C₂²×Q₈

C₂×D₁₃₉

yes

characteristic

C₂²×D₄

C₂×D₁₃₉

yes

characteristic

C₂×M₁₆

—

139

—

C₂×M₁₆

—

139

—

#124

—

#124

—

#124

—

C₂×C₂²⋊C₄

Dic₁₃₉

yes

—

C₂×C₄²

C₂×D₁₃₉

yes

characteristic

C₄⋊C₈

—

278

—

#101

—

278

—

#100

—

278

—

#97

—

278

—

100

C₄×C₈

—

278

—

101

C₂⁴

C₂×Dic₁₃₉

yes

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂²×C₄

C₂×Dic₁₃₉

yes

—

111

C₂²×C₄

C₂×Dic₁₃₉

yes

—

112

C₂²×C₄

C₂²×D₁₃₉

yes

characteristic

113

M₁₆

—

278

—

114

M₁₆

—

278

—

115

M₁₆

—

278

—

116

M₁₆

—

139

—

117

C₂×C₈

—

278

—

118

C₂×C₈

—

278

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

121

C₄²

C₂²⋊D₁₃₉

yes

characteristic

122

C₄²

C₂²⋊D₁₃₉

yes

characteristic

123

C₄²

C₂²⋊D₁₃₉

yes

—

124

C₂³

—

125

C₂³

—

126

C₂³

—

127

C₂³

C₂²×Dic₁₃₉

yes

characteristic

128

Q₈

—

129

Q₈

—

130

Q₈

—

131

D₄

—

132

D₄

—

133

D₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

C₂²⋊Dic₁₃₉

yes

—

138

C₂×C₄

C₂²⋊Dic₁₃₉

yes

—

139

C₂×C₄

D₄×D₁₃₉

yes

—

140

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

characteristic

141

C₂×C₄

C₂×C₂²⋊D₁₃₉

yes

characteristic

142

C₂×C₄

#14555784

yes

—

143

C₈

—

278

—

144

C₈

—

556

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

C₂×C₂²⋊Dic₁₃₉

yes

characteristic

150

C₂²

D₄×Dic₁₃₉

yes

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

#17984813

yes

—

155

C₄

#17984829

yes

characteristic

156

C₄

#17984838

yes

characteristic

157

C₂

—

158

C₂

—

159

C₂

#22580856

yes

characteristic

160

C₂

#22580937

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₁₃₉
commutator	C₂³×C₁₃₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₃₉
Fitting	C₂×#124×C₁₃₉	C₂
socle	C₂²×C₁₃₉	C₂×C₂²⋊C₄
prime radical	C₂²×D₄×C₁₃₉	C₂²
cosocle	C₂²×C₄×C₁₃₉	C₂³
hypercenter	C₂×#124	D₁₃₉
hypocenter	C₁₃₉	#1967
LCS	G ▷ C₂³×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#124

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₁₃₉

commutator

C₂³×C₁₃₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₃₉

Fitting

C₂×#124×C₁₃₉

C₂

socle

C₂²×C₁₃₉

C₂×C₂²⋊C₄

prime radical

C₂²×D₄×C₁₃₉

C₂²

cosocle

C₂²×C₄×C₁₃₉

C₂³

D₁₃₉

C₁₃₉

G ▷ C₂³×C₁₃₉ ▷ C₂×C₁₃₉ ▷ C₁₃₉

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#124

N	H ≅ G/N	type
1. #22580977	C₂	semidirect
2. C₂×#17984761	C₂	semidirect
4. C₄×#14555769	C₂	semidirect
80. C₁₃₉	#1967	semidirect

H ≅ G/N

type

1. #22580977

C₂

semidirect

2. C₂×#17984761

C₂

semidirect

4. C₄×#14555769

C₂

semidirect

80. C₁₃₉

#1967

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22580977	2
C₂×#17984761	2
C₂×#17984759	2
C₄×#14555769	2
C₂×#124×C₁₃₉	2
#1967	139

subgroup

index

C₂×#124×C₁₃₉

139

Subgroups — #31807254

Conjugacy classes of subgroups

Complements

Maximal subgroups